2024春线性代数期中(回忆版)

1.Multiple Choice.Only one choice is correct

1-(1)

Let ${\alpha }_1=\left[\begin{array}{l}2\\ 3\\ 1\end{array}\right],{\alpha }_2=\left[\begin{array}{c}1\\ -1\\ 2\end{array}\right],{\alpha }_3=\left[\begin{array}{l}7\\ 3\\ c\end{array}\right]$

If ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are linearly dependent, then $c$ equals

(A) 5

(B) 6

(C) 7

(D) 8

1-(2)

Let $A$ be an $m\times n$ real matrix and $b$ be an $m\times 1$ real column vector.Which of the following statements is correct?

(A) If $Ax=b$ does not have any solution, then $Ax=0$ has only the zero solution

(B) If $Ax=0$ has infinitely many solutions, then $Ax=b$ has infinitely many solutions

(C) If $m<n$, both $Ax=b$ and $Ax=0$ have infinitely many solutions

(D) If the rank of $A$ is $n$, then $Ax=0$ has only the zero solution

1-(3)

For which value of $k$ does the system

$$\{\begin{array}{l}{x}_1+2x_2-4x_3+3x_4=0\\ x_1+3x_2-2x_3-2x_4=0\\ x_1+5x_2+(5-k)x_3-12x_4=0\end{array}$$

have exactly two free variables?

(A) 5

(B) 4

(C) 3

(D) 2

1-(4)

Let $u,v\in R^3$ and $\lambda \in R$ .Which of the following statement is false?

(A) If $u$ and $v$ are nonzero vectors satisfying $u^{\mathrm{\top }}v=0$, then $u$ and $v$ are linearly independent

(B) If $u+v$ is orthogonal to $u-v$, then $\parallel u\parallel =\parallel v\parallel$

(C) $u^{\mathrm{\top }}v=0$ if and only if $u=0$ or $v=0$

(D) $\lambda \nu =0$ if and only if $v=0$ or $\lambda =0$

1-(5)

Let $A$ and $B$ be two $n\times n$ matrices.Which of the following assertions is false?

(A) If $A,B$ are symmetric matrices, then $AB$ is a symmetric matrix

(B) If $A,B$ are invertible matrices, then $AB$ is an invertible matrix

(C) If $A,B$ are permutation matrices, then $AB$ is a permutation matrix

(D) If $A,B$ are upper triangular matrices, then $AB$ is an upper triangular matrix

2.Fill in the blanks

2-(1)

Let $A=\left[\begin{array}{lll}1& 0& 0\\ a& 1& 0\\ b& 3& 2\end{array}\right],a,b\in R$

Then $A^{-1}=\underset{―}{}$

2-(2)

Let A be a $4\times 3$ real matrix with rank 2 and B = $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 0\\ -1& 0& 3\end{array}\right]$, the rank of AB is $\underset{―}{}$

2-(3)

Let Let $A=\left[\begin{array}{ccc}1& -1& 1\\ -1& 1& -1\\ 2& -2& 2\end{array}\right]$. Then $A^{2024}=$ $\underset{―}{}$

2-(4)

Consider the system of linear equations:

$$Ax=b:\{\begin{array}{l}x=2\\ y=3\\ x+y=6\end{array}$$

The least-squares solution for the system is $\underset{―}{}$

3

Let

$$A=\left[\begin{array}{ccc}1& -2& 3\\ 2& -5& 1\\ 1& -4& -7\end{array}\right]$$

Find an LU factorization of $A$

4

Consider the following $4\times 5$ matrix $A$ and 4 -dimensional column vector $b$ :

$$A=\left[\begin{array}{ccccc}0& 2& 4& 1& 6\\ 0& 1& 1& 1& 3\\ 0& 4& 10& 1& 2\\ 0& -1& -5& 1& 7\end{array}\right],b=\left[\begin{array}{c}3\\ 2\\ -5\\ 10\end{array}\right]$$

(A) Find a basis for each of the four fundamental subspace of $A$

(B) Find the complete solution to $Ax=b$

5

Let $A=\left[\begin{array}{ll}1& 1\\ 0& 2\end{array}\right]$ and $T$ be the linear transformation from $R^{2\times 2}$ to $R^{2\times 2}$ defined

$$\text{ by }T(x)=XA+AX,X\in R^{2\times 2}$$

where $R^{2\times 2}$ denotes the vector space consisting of all $2\times 2$ real matrices

(A) Find the matrix representation of $T$ with respect to the following ordered basis

$$v_1=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],v_2=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],v_3=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],v_4=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],$$

(B) Find a matrix $B$ such that

$$\begin{array}{c}T(B)=\left[\begin{array}{ll}0& 0\\ 0& 0\end{array}\right]\end{array}$$

(C) Find a matrix $C$ such that

$$\begin{array}{rl}& T(C)=\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]\end{array}$$

6

Let $A,B$ be two $n\times n$ real matrices satisfying $A^2=A$ and $B^2=B$. Show that if $(A+B{)}^2=A+B$, then $AB=0$. When 0 denotes the $n\times n$ zero matrix

7

Let $A$ be a $3\times 2$ matrix, $B$ be a $2\times 3$ matrix such that

$$AB=\left[\begin{array}{ccc}8& 0& -4\\ -\frac{3}{2}& 9& -6\\ -2& 0& 1\end{array}\right]$$

(A) Compute $(AB{)}^2$

(B) Find BA